Quantum Fisica

January 19, 2009 10:44 WSPC/187-IJQI 00483

International Journal of Quantum InformationVol. 7, Supplement (2009) 125–137c! World Scientific Publishing Company

QUANTUM ESTIMATION FOR QUANTUM TECHNOLOGY

MATTEO G. A. PARIS

Dipartimento di Fisica dell’Universita di Milano,I-20133 Milano, Italia

CNSIM, Udr Milano, I-20133 Milano, Italia

ISI Foundation, I-10133 Torino, Italia

Received 12 November 2008

Several quantities of interest in quantum information, including entanglement and purity,are nonlinear functions of the density matrix and cannot, even in principle, correspondto proper quantum observables. Any method aimed to determine the value of thesequantities should resort to indirect measurements and thus corresponds to a parameterestimation problem whose solution, i.e. the determination of the most precise estimator,unavoidably involves an optimization procedure. We review local quantum estimationtheory and present explicit formulas for the symmetric logarithmic derivative and thequantum Fisher information of relevant families of quantum states. Estimability of aparameter is defined in terms of the quantum signal-to-noise ratio and the number ofmeasurements needed to achieve a given relative error. The connections between theoptmization procedure and the geometry of quantum statistical models are discussed.Our analysis allows to quantify quantum noise in the measurements of non observablequantities and provides a tools for the characterization of signals and devices in quantumtechnology.

Keywords: Quantum estimation; Fisher information.

1. Introduction

Many quantities of interest in physics are not directly accessible, either in principleor due to experimental impediments. This is particolarly true for quantum mechan-ical systems where relevant quantities like entanglement and purity are nonlinearfunctions of the density matrix and cannot, even in principle, correspond to properquantum observables. In these situations one should resort to indirect measure-ments, inferring the value of the quantity of interest by inspecting a set of datacoming from the measurement of a di!erent obeservable, or a set of observables.This is basically a parameter estimation problem which may be properly addressedin the framework of quantum estimation theory (QET),1 which provides analyticaltools to find the optimal measurement according to some given criterion. In turn,there are two main paradigms in QET: Global QET looks for the POVM minimizinga suitable cost functional, averaged over all possible values of the parameter to beestimated. The result of a global optimization is thus a single POVM, independent

125


126 M. G. A. Paris

on the value of the parameter. On the other hand, local QET looks for the POVMmaximizing the Fisher information, thus minimizing the variance of the estima-tor, at a fixed value of the parameter.2–6 Roughly speaking, one may expect localQET to provide better performances since the optimization concerns a specificvalue of the parameter, with some adaptive or feedback mechanism assuring theachievability of the ultimate bound.7 Global QET has been mostly applied to findoptimal measurements and to evaluate lower bounds on precision for the estima-tion of parameters imposed by unitary transformations. For bosonic systems theseinclude single-mode phase,8,9 displacement,10 squeezing 11,12 as well as two-modetransformations, e.g. bilinear coupling.13 Local QET has been applied to the esti-mation of quantum phase14 and to estimation problems with open quantum systemsand non unitary processes15: to finite dimensional systems,16 to optimally estimatethe noise parameter of depolarizing17 or amplitude-damping,18 and for continuousvariable systems to estimate the loss parameter of a quantum channel19–22 as wellas the position of a single photon.23 Recently, the geometric structure induced bythe Fisher information itself has been exploited to give a quantitative operationalinterpretation for multipartite entanglement24 and to assess quantum criticality asa resource for quantum estimation.25

In this paper we review local quantum estimation theory and present explicitformulas for the symmetric logarithmic derivative and the quantum Fisher infor-mation of relevant families of quantum states. We are interested in evaluating theultimate bound on precision (sensitivity), i.e. the smallest value of the parameterthat can be discriminated, and to determine the optimal measurement achievingthose bounds. Estimability of a parameter will be then defined in terms of thequantum signal-to-noise ratio and the number of measurements needed to achievea given relative error.

The paper is structured as follows. In the next Section we review local quan-tum estimation theory and report the solution of the optimization problem, i.e. thedetermination of the optimal quantum estimator in terms of the symmetric logarith-mic derivative, as well as the ultimate bounds to precision in terms of the quantumFisher information. General formulas for the symmetric logarithmic derivative andthe quantum Fisher information are derived. In Sec. 3 we address the quantificationof estimability of a parameter put forward the quantum signal-to-noise ratio andthe number of measurements needed to achieve a given relative error as the suitablefigures of merit. In Sec. 4 we present explicit formulas for sets of pure states and thegeneric unitary family. We also consider the multiparamer case and the problem ofrepametrization. In Sec. 5 we discuss the connections between estimability of a setof parameters, the optmization procedure and the geometry of quantum statisticalmodels. Sec. 6 closes the paper with some concluding remarks.

2. Local Quantum Estimation Theory

The solution of a parameter estimation problem amounts to find an estimator, i.e.a mapping ! = !(x1, x2, . . .) from the set " of measurement outcomes into the


Local Quantum Estimation 127

space of parameters. Optimal estimators in classical estimation theory are thosesaturating the Cramer-Rao inequality26

V(!) ! 1MF (!)

(1)

which establishes a lower bound on the mean square error V (!) = E![(!({x})"!)2]of any estimator of the parameter !. In Eq. (1) M is the number of measurementsand F (!) is the so-called Fisher Information (FI)

F (!) =!

dxp(x|!)"# ln p(x|!)

#!

#2

=!

dx1

p(x|!)

"#p(x|!)

#!

#2

. (2)

where p(x|!) denotes the conditional probability of obtaining the value x when theparameter has the value !. For unbiased estimators, as those we will deal with, themean square error is equal to the variance Var(!) = E![!2] " E![!]2.

When quantum systems are involved any estimation problem may be stated byconsidering a family of quantum states $! which are defined on a given Hilbertspace H and labeled by a parameter ! living on a d-dimensional manifold M, withthe mapping ! #$ $! providing a coordinate system. This is sometimes referredto as a quantum statistical model. The parameter ! does not, in general, corre-spond to a quantum observable and our aim is to estimate its values through themeasurement of some observable on $!. In turn, a quantum estimator O! for !is a selfadjoint operator, which describe a quantum measurement followed by anyclassical data processing performed on the outcomes. The indirect procedure ofparameter estimation implies an additional uncertainty for the measured value,that cannot be avoided even in optimal conditions. The aim of quantum estima-tion theory is to optimize the inference procedure by minimizing this additionaluncertainty.

In quantum mechanics, according to the Born rule we have p(x|!) = Tr["x$!]where {"x},

$dx"x = I, are the elements of a positive operator-valued measure

(POVM) and $! is the density operator parametrized by the quantity we wantto estimate. Introducing the Symmetric Logarithmic Derivative (SLD) L! as theselfadjoint operator satistying the equation

L!$! + $!L!

2=

#$!

#!(3)

we have that #!p(x|!) = Tr[#!$!"x] = Re(Tr[$!"xL!]). The Fisher Information(2) is then rewritten as

F (!) =!

dxRe (Tr [$!"xL!])2

Tr[$!"x]. (4)

For a given quantum measurement, i.e. a POVM {"x}, Eqs. (2) and (4) establish theclassical bound on precision, which may be achieved by a proper data processing,e.g. by maximum likelihood, which is known to provide an asymptotically e#cientestimator. On the other hand, in order to evaluate the ultimate bounds to precision


128 M. G. A. Paris

we have now to maximize the Fisher information over the quantum measurements.Following Refs. 3–6 we have

F (!) %!

dx

%%%%%Tr [$!"xL!]&

Tr[$!"x]

%%%%%

2

(5)

=!

dx

%%%%%Tr

' &$!

&"x&

Tr [$!"x]

&"xL!

&$!

(%%%%%

2

%!

dxTr ["xL!$!L!] (6)

= Tr[L!$!L!]

= Tr[$!L2!]

The above chain of inequalities prove that the Fisher information F (!) of any quan-tum measurement is bounded by the so-called Quantum Fisher Information (QFI)

F (!) % H(!) ' Tr[$!L2!] = Tr[#!$!L!] (7)

leading the quantum Cramer-Rao bound

Var(!) ! 1MH(!)

(8)

to the variance of any estimator. The quantum version of the Cramer-Rao theo-rem provides an ultimate bound: it does depend on the geometrical structure ofthe quantum statistical model and does not depend on the measurement. Opti-mal quantum measurements for the estimation of ! thus corresponds to POVMwith a Fisher information equal to the quantum Fisher information, i.e. those sat-urating both inequalities (5) and (6). The first one is saturated when Tr[$!"xL!]is a real number (!. On the other hand, Ineq. (6) is based on the Schwartzinequality |Tr[A†B]|2 % Tr[A†A]Tr[B†B] applied to A† = &

$!&"x/

&Tr[$!"x]

and B =&"xL!

&$! and it is saturated when

&"x

&$!

Tr [$!"x]=

&"xL!

&$!

Tr[$!"xL!](!, (9)

The operatorial condition in Eq. (9) is satisfied i! {"x} is made by the set of pro-jectors over the eigenstates of L!, which, in turn, represents the optimal POVMto estimate the parameter !. Notice, however, that L! itself may not represent theoptimal observable to be measured. In fact, Eq. (9) determines the POVM andnot the estimator i.e. the function of the eigenvalues of L!. As we have alreadymentioned above, this corresponds to a classical post-processing of data aimed tosaturate the Cramer-Rao inequality (1) and may be pursued by maximum likeli-hood, which is known to provide an asymptotically e#cient estimator. Using thefact that Tr[$!L!] = 0 an explicit form for the optimal quantum estimator is



given by

O! = !I +L!

H(!)(10)

for which we have

Tr[$!O!] = !, Tr[$!O2!] = !2 +

Tr[$!L2!]

H2(!), and thus )$O2

!* = 1/H(!).

Equation (3) is Lyapunov matrix equation to be solved for the SLD L!. Thegeneral solution may be written as

L! = 2! !

0dt exp{"$!t} #!$! exp{"$!t} (11)

which, upon writing $! in its eigenbasis $! =)

n $n|%n*)%n|, leads to

L! = 2*

nm

)%m|#!$!|%n*$n + $m

|%m*)%n|, (12)

where the sums include only terms with $n + $m += 0. The quantum Fisher infor-mation is thus given by

H(!) = 2*

nm

|)%m|#!$!|%n*|2

$n + $m, (13)

or, in a basis independent form,

H(!) = 2! !

0dt Tr[#!$! exp{"$!t} #!$! exp{"$!t}]. (14)

Notice that the SLD is defined only on the support of $! and that both the eigen-values $n and the eigenvectors |%n* may depend on the parameter. In order toseparate the two contribution to the QFI we explicitly evaluate #!$!

#!$! =*

p

#!$p|%p*)%p| + $p|#!%p*)%p| + $p|%p*)#!%p| (15)

The symbol |#!%n* denotes the ket |#!%n* =)

k #!%nk|k*, where %nk are obtainedexpanding |%n* in arbitrary basis {|k*} independent on !. Since )%n|%m* = &nm wehave #!)%n|%m* ' )#!%n|%m* + )%n|#!%m* = 0 and therefore

Re)#!%n|%m* = 0 )#!%n|%m* = ")%n|#!%m* = 0.

Using Eq. (15) and the above identities we have

L! =*

p

#!$p

$p|%p*)%p| + 2

*

n"=m

$n " $m

$n + $m)%m|#!%n*|%m*)%n| (16)


130 M. G. A. Paris

and in turn

H(!) =*

p

(#!$p)2

$p+ 2

*

n"=m

'nm |)%m|#!%n*|2 (17)

where

'nm =($n " $m)2

$n + $m+ any antisymmetric term, (18)

as for example

'nm = 2$n$n " $m

$n + $m'nm = 2$n

"$n " $m

$n + $m

#2

(19)

The first term in Eq. (17) represents the classical Fisher information of the dis-tribution {$p} whereas the second term contains the truly quantum contribution.The second term vanishes when the eigenvectors of $! do not depend. In this case[$!, #!$!] = 0 and Eq. (11) reduces to L! = #! log $!.

Finally, upon substituting the above Eqs. in Eq. (10), we obtain the correspond-ing optimal quantum estimator

O! =*

p

"! +

#!$p

$p

#|%p*)%p| +

2H(!)

*

n"=m

$n " $m

$n + $m)%m|#!%n*|%m*)%n|. (20)

So far we have considered the case of a parameter with a fixed given value. Aquestion arises on whether a bound for estimator variance may be established alsofor a parameter having an a priori distribution z(!). The answer is positive andgiven by the Van Trees inequality28,29 which provides a bound for the averagevariance

Var(!) =!

dx

!d!z(!)[!({x}) " !)]2

of any unbiased estimator of the random parameter !. Van Trees inequality statesthat

Var(!) ! 1ZF

(21)

where the generalized Fisher information ZF is given by

ZF =!

dx

!d!p(x,!) [#! log p(x,!)]2 , (22)

p(x,!) being the joint probability distribution of the outcomes and the parameterof interest. Upon writing the joint distribution as p(x,!) = p(x|!)z(!) Eq. (22)may be rewritten as

ZF =!

d!z(!)F (!) + M

!d!z(!)[#! log z(!)]2. (23)

Equation (23) says that the generalized Fisher information is the sum of two terms,the first is simply the average of the Fisher information over the a priori distribution



whereas the second term is the Fisher information of the priori distribution itself. Asexpected, in the asymptotic limit of many measurements the a priori distribution isno longer relevant. The quantity ZF is upper bounded by the analogue expressionZH where the average of the Fisher information is replaced by the average of theQFI H(!) The resulting quantum Van Trees bound may be easily written as

Var(!) ! 1ZH

. (24)

3. Estimability of a Parameter

A large signal is easily estimated whereas a quantity with a vanishing value may beinferred only if the corresponding estimator is very precise i.e. characterized by asmall variance. This intuitive statement indicates that in assessing the performancesof an estimator and, in turn, the overall estimability of a parameter, the relevantfigure of merit is the scaling of the variance with the mean value rather than itsabsolute value. This feature may be quantified by means of the signal-to-noise ratio(for a single measurement)

R! =!2

Var(!)which is larger for better estimators. Using the quantum Cramer-Rao bound oneeasily derives that the signal-to-noise ratio of any estimator is bounded by thequantity

R! % Q! ' !2H(!)

which we refer to as the quantum signal-to-noise ratio. We say that a given param-eter ! is e!ectively estimable quantum-mechanically when the corresponding Q! islarge.

Upon taking into account repeated measurements we have that the number ofmeasurements leading to a 99.9% (3') confidence interval corresponds to a relativeerror

&2 =9Var(!)M!2

=9M

1Q!

=9

M!2H(!)Therefore, the number of measurements needed to achieve a 99.9% confidence inter-val with a relative error & scales as

M" =9&2

1Q!

In other words, a vanishing Q! implies a diverging number of measurements toachieve a given relative error, whereas a finite value allows estimation with arbitraryprecision at finite number of measurements.

4. Examples

In this section we provide explicit evaluation of the symmetric logarithmic deriva-tive and the quantum Fisher information for relevant families of quantum states,


132 M. G. A. Paris

including sets of pure states and the generic unitary family. We also consider themultiparameter case and the problem of repametrization.

4.1. Unitary families and the pure state model

Let us consider the case where the parameter of interest is the amplitude of a unitaryperturbation imposed to a given initial state $0. The family of quantum states we aredealing with may be expressed as $! = U!$0U

†! where U! = exp{"i!G} is a unitary

operator and G is the corresponding Hermitian generator. Upon expanding theunperturbed state in its eigenbasis $0 =

)$n|(n*)(n| we have $! =

)n $n|%n*)%n|

where |%n* = U!|(n*. As a consequence we have

#!$! = iU![G, $0]U †!.

and the SLD is may be written as L! = U!L0U†! where L0 is given by

L0 = 2i*

n,m

)(m|[G, $0]|(n*$n + $m

|(n*)(m|

= 2i*

n"=m

)(m|G|(n*$n " $m

$n + $m|(n*)(m|. (25)

The corresponding quantum Fisher information is independent on the value ofparameter and may be written in compact form as

H = Tr[$0 L20] = Tr[$0 [L0, G]] = Tr[L0 [G, $0]] = Tr[G [$0, L0]]

or, more explicitly, as

H = 2*

n"=m

'nmG2nm

where the elements 'nm are given in Eq. (18), or equivalently (19), and Gnm =)(n|G|(m* = )%n|G|%m* denote the matrix element of the generator G in eitherthe eigenbasis of $0 or $!.

For a generic family of pure states we have $! = |%!*)%!|. Since $2! = $! we

have #!$! = #!$! $! + $!#!$! and thus L! = 2#!$! = |%!*)#!%!| + |#!%!*)%!|.Finally we have

H(!) = 4[)#!%!|#!%!* + ()#!%!|%!*)2] (26)

For a unitary family of pure states |%!* = U!|%0* we have

|#!%!* = "iGU!|%0* = "iG|%!*,)#!%!|#!%!* = )%0|G2|%0*,

)#!%!|%!* = "i)%0|G|%0*.



The quantum Fisher information thus reduces to the simple form

H = 4)%0|$G2|%0* (27)

which is independent on ! and proportional to the fluctuations of the generatoron the unperturbed state. Using Eq. (27) the quantum Cramer-Rao bound in (8)rewrites in the appealing form27

Var(!))$G2* ! 14M

, (28)

which represents a parameter-based uncertainty relation which applies also whenthe shift parameter ! in the unitary U! = e#i!G does not correspond to the observ-able canonically conjugate to G. When the unperturbed state is not pure the QFImay be written as

H = 4 Tr[$G2$0] + 4*

n

$n)(n|)G*2 " 2GK(n)G|(n* (29)

K(n) =*

m

$m

$n + $m|(m*)(m| #0$|$0%&$0|"$ 1

2|(0*)(0| (30)

and Eq. (28) becomes

Var(!))$G2* ! 14M

'1 +

*

n

$n)(n|)G*2 " 2GK(n)G|(n*(#1

. (31)

The second term in Eqs. (29) and (31) thus represents the classical contribution touncertainty due to the mixing of the initial signal.

As we have seen, for unitary families of quantum states the QFI is independenton the value of the parameter. As a consequence the quantum signal-to-noise ratioQ! vanishes for vanishing ! and thus the number of measurements needed to achievea relative error & diverges as M" , (&!)#2.

4.2. Quantum operations

Let us now consider a family of quantum states obtained from a given inital state$0 by the action of a generic quantum operation $! = E!($0) =

)k Mk!$0M

†k!.

Upon writing the initial and the evolved states in terms of their eigenbasis $0 =)s $0s|(s*)(s|, $! =

)s $n|%n*)%n| we may evaluate the SLD and the quantum

Fisher information using Eqs. (12) and (13) where

$n =*

ks

$0s|)%n|Mk!|(s*|2 (32)

)%m|#!$!|%n* =*

ks

$0s[)%m|#!Mk!|(s*)(s|M †k!|%n*

+ )%m|Mk!|(s*)(s|#!M †k!|%n*]. (33)

For a pure state at the input $0 = |%0*)%0| the above equation rewrites withoutthe sum over s.


134 M. G. A. Paris

4.3. Multiparametric models and reparametrization

In situations where more than one parameter is involved, the family of quantumstates $! depends on a set ! = {!µ}, µ = 1, . . . , N . In this cases the relevant objectin the estimation problem is given by the so-called quantum Fisher informationmatrix, whose elements are defined as

H(!)µ% = Tr+$!

LµL% + L%Lµ

2

,= Tr[#%$!Lµ] = Tr[#µ$!L% ]

=*

n

(#µ$n)(#%$n)$n

+*

n"=m

($n " $m)2

$n + $m

- [)%n|#µ%m*)#%%m|%n* + )%n|#%%m*)#µ%m|%n*] (34)

where Lµ is the SLD corresponding to the parameter !µ. The Cramer-Rao theoremfor multiparameter estimation says that the inverse of the Fisher matrix providesa lower bound on the covariance matrix Cov["]ij = )!i!j* " )!i*)!j*, i.e.

Cov["] ! 1M

H(!)#1

The above relation is a matric inequality and the corresponding bound may notbe achievable achievable in a multiparameter estimation. On the other hand, thediagonal elements of the inverse Fisher matrix provide achievable bounds for thevariances of single parameter estimators at fixed value of the others, in formula

Var(!µ) = )µµ ! 1M

(H#1)µµ. (35)

Of course, for a diagonal Fisher matrix Var(!µ) ! 1/Hµµ.Let us now suppose that the quantity of interest g is a known function g(!) of the

parameters used to label the family of states. In this case we need to reparametrizethe familiy with a new set of parameters -! = {-!j = -!j(!) that includes the quantityof interest, e.g. -!1 ' g(!). Since -#µ =

)% Bµ%#% where Bµ% = #!%/#-!µ it is easy

to prove that

-Lµ =*

%

Bµ%L%.H = BHBT .

The ultimate precision on the estimation of g at fixed values of the other parametersis thus given by

Var(g) ! 1M

(.H#1

)11

5. Geometry of Quantum Estimation

The estimability of a set of parameters labelling the family of quantum states{$!} is naturally related to the distinguishability of the states within the quantumstatistical model i.e. with the notions of distance. On the manifold of quantumstates, however, di!erent distances may be defined and a question arises on which of



them captures the notion of estimation measure. As it can be easily proved it turnsout that the Bures distance30–36 is the proper quantity to be taken into account.This may be seen as follows. The Bures distance between two density matrices isdefined as D2

B($,') = 2[1 "&

F ($,')] where F ($,') = (Tr[&&

$'&$])2 is the

fidelity. The Bures metric gµ% is obtained upon considering the distance for twostates obtained by an infinitesimal change in the value of the parameter

d2B = D2

B($!, $!+d!) = gµ%d!µd!% .

By explicitly evaluating the Bures distance37 one arrives at gµ% = 1/4Hµ%(!),i.e. the Bures metric is simply proportional to the QFI, which itself is symmetric,real and positive semidefinite, i.e. represents a metric for the manifold underlyingthe quantum statistical model. Indeed, a large QFI for a given ! implies that thequantum states $! and $!+d! should be statistically distinguishable more e!ectivelythan the analogue states for a value ! corresponding to smaller QFI. In other words,one confirms the intuitive picture in which optimal estimability (that is, a divergingQFI) corresponds to quantum states that are sent far apart upon infinitesimalvariations of the parameters.

The structures described above are pictorially described in Fig. 1. The ideais that any measurement aimed to estimate the parameters ! turns the set ofparameters into a statistical di!erential manifold endowed with the Fisher metricF µ%(!). On the other hand, when the parameters are mapped into the manifold ofquantum states the statistical distance is expressed in terms of the Bures metric.The connection between the two constructions is provided by the optimization of theestimation procedure over quantum measurements, which shows that the Quantum

Fig. 1. Geometry of quantum estimation.


136 M. G. A. Paris

Fisher metric Hµ%(!) is the bound to F µ%(!) and coincides, apart from a factorfour, with the Bures metric.

6. Conclusion and Outlook

As a matter of fact, there are many quantities of interest that do not correspondto any quantum observable. Among these, we mention the amount of entanglementand the purity of a quantum state and the coupling constant of an interactionHamiltonian or a quantum operation. In these situations, the values of the quantityof interest can be indirectly inferred by an estimation procedure, i.e. by measuringone or more proper observables, a quantum estimator, and then manipulating theoutcomes by a suitable classical processing.

In this paper, upon exploiting the geometric theory of quantum estimation,we have described a general method to solve a quantum statistical model, i.e. tofind the optimal quantum estimator and to evaluate the corresponding bounds toprecision. To this aim we used the quantum Cramer-Rao theorem and the explicitevaluation of the quantum Fisher information matrix. We have derived the explicitform of the optimal observable in terms of the symmetric logarithmic derivativeand evaluated the corresponding bounds to precision, which represent the ultimatebound posed by quantum mechanics to the precision of parameter estimation. Forunitary families of quantum states the bounds may expressed in the form of aparameter-based uncertainty relation.

The analysis reported in this paper has a fundamental interest and represents arelevant tool in the design of realistic quantum information protocols. The approachhere outlined is currently being applied to the estimation of entanglement38 andthe coupling constant of an interaction Hamiltonian.25,39

Acknowledgments

The author thanks Paolo Giorda, Alex Monras, Paolo Zanardi, Marco Genoni,Michael Korbman, Carmen Invernizzi and Stefano Olivares for stimulatingdiscussions.

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